Bounds of the incremental gain for discrete-time recurrent neural networks

As a nonlinear system, a recurrent neural network generally has an incremental gain different from its induced norm. While most of the previous research efforts were focused on the latter, this paper presents a method to compute an effective upper bound of the former for a class of discrete-time recurrent neural networks, which is not only applied to systems with arbitrary inputs but also extended to systems with small-norm inputs. The upper bound is computed by simple optimizations subject to linear matrix inequalities (LMIs). To demonstrate the wide connections of our results to problems in control, the servomechanism is then studied, where a feedforward neural network is designed to control the output of a recurrent neural network to track a set of trajectories. This problem can be converted into the synthesis of feedforward-feedback gains such that the incremental gain of a certain system is minimized. An algorithm to perform such a synthesis is proposed and illustrated with a numerical example.